Log-majorizations between quasi-geometric type means for matrices
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909995046535168 |
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| author | Hiai, Fumio |
| author_facet | Hiai, Fumio |
| contents | In this paper, for $α\in(0,\infty)\setminus\{1\}$, $p>0$ and positive semidefinite matrices $A$ and $B$, we consider the quasi-extension $\mathcal{M}_{α,p}(A,B):=\mathcal{M}_α(A^p,B^p)^{1/p}$ of several $α$-weighted geometric type matrix means $\mathcal{M}_α(A,B)$ such as the $α$-weighted geometric mean in Kubo--Ando's sense, the Rényi mean, etc. The log-majorization $\mathcal{M}_{α,p}(A,B)\prec_{\log}\mathcal{N}_{α,q}(A,B)$ is examined for pairs $(\mathcal{M},\mathcal{N})$ of those $α$-weighted geometric type means. The joint concavity/convexity of the trace functions $\mathrm{Tr}\,\mathcal{M}_{α,p}$ is also discussed based on theory of quantum divergences. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_04691 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Log-majorizations between quasi-geometric type means for matrices Hiai, Fumio Functional Analysis Quantum Physics 15A45, 47A64 In this paper, for $α\in(0,\infty)\setminus\{1\}$, $p>0$ and positive semidefinite matrices $A$ and $B$, we consider the quasi-extension $\mathcal{M}_{α,p}(A,B):=\mathcal{M}_α(A^p,B^p)^{1/p}$ of several $α$-weighted geometric type matrix means $\mathcal{M}_α(A,B)$ such as the $α$-weighted geometric mean in Kubo--Ando's sense, the Rényi mean, etc. The log-majorization $\mathcal{M}_{α,p}(A,B)\prec_{\log}\mathcal{N}_{α,q}(A,B)$ is examined for pairs $(\mathcal{M},\mathcal{N})$ of those $α$-weighted geometric type means. The joint concavity/convexity of the trace functions $\mathrm{Tr}\,\mathcal{M}_{α,p}$ is also discussed based on theory of quantum divergences. |
| title | Log-majorizations between quasi-geometric type means for matrices |
| topic | Functional Analysis Quantum Physics 15A45, 47A64 |
| url | https://arxiv.org/abs/2510.04691 |